1 //===-- Single-precision atan function ------------------------------------===// 2 // 3 // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions. 4 // See https://llvm.org/LICENSE.txt for license information. 5 // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception 6 // 7 //===----------------------------------------------------------------------===// 8 9 #include "src/math/atanf.h" 10 #include "inv_trigf_utils.h" 11 #include "src/__support/FPUtil/FPBits.h" 12 #include "src/__support/FPUtil/PolyEval.h" 13 #include "src/__support/FPUtil/except_value_utils.h" 14 #include "src/__support/FPUtil/multiply_add.h" 15 #include "src/__support/FPUtil/nearest_integer.h" 16 #include "src/__support/FPUtil/rounding_mode.h" 17 #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY 18 19 namespace LIBC_NAMESPACE { 20 21 LLVM_LIBC_FUNCTION(float, atanf, (float x)) { 22 using FPBits = typename fputil::FPBits<float>; 23 24 constexpr double FINAL_SIGN[2] = {1.0, -1.0}; 25 constexpr double SIGNED_PI_OVER_2[2] = {0x1.921fb54442d18p0, 26 -0x1.921fb54442d18p0}; 27 28 FPBits x_bits(x); 29 Sign sign = x_bits.sign(); 30 x_bits.set_sign(Sign::POS); 31 uint32_t x_abs = x_bits.uintval(); 32 33 // x is inf or nan, |x| < 2^-4 or |x|= > 16. 34 if (LIBC_UNLIKELY(x_abs <= 0x3d80'0000U || x_abs >= 0x4180'0000U)) { 35 double x_d = static_cast<double>(x); 36 double const_term = 0.0; 37 if (LIBC_UNLIKELY(x_abs >= 0x4180'0000)) { 38 // atan(+-Inf) = +-pi/2. 39 if (x_bits.is_inf()) { 40 volatile double sign_pi_over_2 = SIGNED_PI_OVER_2[sign.is_neg()]; 41 return static_cast<float>(sign_pi_over_2); 42 } 43 if (x_bits.is_nan()) 44 return x; 45 // x >= 16 46 x_d = -1.0 / x_d; 47 const_term = SIGNED_PI_OVER_2[sign.is_neg()]; 48 } 49 // 0 <= x < 1/16; 50 if (LIBC_UNLIKELY(x_bits.is_zero())) 51 return x; 52 // x <= 2^-12; 53 if (LIBC_UNLIKELY(x_abs < 0x3980'0000)) { 54 #if defined(LIBC_TARGET_CPU_HAS_FMA) 55 return fputil::multiply_add(x, -0x1.0p-25f, x); 56 #else 57 double x_d = static_cast<double>(x); 58 return static_cast<float>(fputil::multiply_add(x_d, -0x1.0p-25, x_d)); 59 #endif // LIBC_TARGET_CPU_HAS_FMA 60 } 61 // Use Taylor polynomial: 62 // atan(x) ~ x * (1 - x^2 / 3 + x^4 / 5 - x^6 / 7 + x^8 / 9 - x^10 / 11). 63 constexpr double ATAN_TAYLOR[6] = { 64 0x1.0000000000000p+0, -0x1.5555555555555p-2, 0x1.999999999999ap-3, 65 -0x1.2492492492492p-3, 0x1.c71c71c71c71cp-4, -0x1.745d1745d1746p-4, 66 }; 67 double x2 = x_d * x_d; 68 double x4 = x2 * x2; 69 double c0 = fputil::multiply_add(x2, ATAN_TAYLOR[1], ATAN_TAYLOR[0]); 70 double c1 = fputil::multiply_add(x2, ATAN_TAYLOR[3], ATAN_TAYLOR[2]); 71 double c2 = fputil::multiply_add(x2, ATAN_TAYLOR[5], ATAN_TAYLOR[4]); 72 double p = fputil::polyeval(x4, c0, c1, c2); 73 double r = fputil::multiply_add(x_d, p, const_term); 74 return static_cast<float>(r); 75 } 76 77 // Range reduction steps: 78 // 1) atan(x) = sign(x) * atan(|x|) 79 // 2) If |x| > 1, atan(|x|) = pi/2 - atan(1/|x|) 80 // 3) For 1/16 < x <= 1, we find k such that: |x - k/16| <= 1/32. 81 // 4) Then we use polynomial approximation: 82 // atan(x) ~ atan((k/16) + (x - (k/16)) * Q(x - k/16) 83 // = P(x - k/16) 84 double x_d, const_term, final_sign; 85 int idx; 86 87 if (x_abs > 0x3f80'0000U) { 88 // |x| > 1, we need to invert x, so we will perform range reduction in 89 // double precision. 90 x_d = 1.0 / static_cast<double>(x_bits.get_val()); 91 double k_d = fputil::nearest_integer(x_d * 0x1.0p4); 92 x_d = fputil::multiply_add(k_d, -0x1.0p-4, x_d); 93 idx = static_cast<int>(k_d); 94 final_sign = FINAL_SIGN[sign.is_pos()]; 95 // Adjust constant term of the polynomial by +- pi/2. 96 const_term = fputil::multiply_add(final_sign, ATAN_COEFFS[idx][0], 97 SIGNED_PI_OVER_2[sign.is_neg()]); 98 } else { 99 // Exceptional value: 100 if (LIBC_UNLIKELY(x_abs == 0x3d8d'6b23U)) { // |x| = 0x1.1ad646p-4 101 return sign.is_pos() ? fputil::round_result_slightly_down(0x1.1a6386p-4f) 102 : fputil::round_result_slightly_up(-0x1.1a6386p-4f); 103 } 104 // Perform range reduction in single precision. 105 float x_f = x_bits.get_val(); 106 float k_f = fputil::nearest_integer(x_f * 0x1.0p4f); 107 x_f = fputil::multiply_add(k_f, -0x1.0p-4f, x_f); 108 x_d = static_cast<double>(x_f); 109 idx = static_cast<int>(k_f); 110 final_sign = FINAL_SIGN[sign.is_neg()]; 111 const_term = final_sign * ATAN_COEFFS[idx][0]; 112 } 113 114 double p = atan_eval(x_d, idx); 115 double r = fputil::multiply_add(final_sign * x_d, p, const_term); 116 117 return static_cast<float>(r); 118 } 119 120 } // namespace LIBC_NAMESPACE 121